Politically the Romans during the medieval period and the Romans of the Republic had a lot of differences but similarities. A lot of Republican institutions (I.e. the Senate) stayed in tact.

In Anthony Kaldellis' book "The Byzantine Republic" he argues that the res publica or the politea never ceased to exist. In order for an emperor to be crowned in Constantinople they needed the support of the people, in 511 Anastasios had a dispute with the Patriarch Makedonios. The citizens of Constantinople showed support of the Patriarch forcing the Emperor to bring him back into his palace circle. This demonstrated that the Emperor wasn't in power unless the people supported him (a quasi vote of confidence).

In the case of Justin and Justinian, it is recorded that there were back room dealings with prominent senators in order to influence the politea to allow Justin to accend to the throne. Justinian also struggled to keep his power when the people and the Senate attempted to usurp his rule during the Nika Riots. The Senate instigated raising other emperor's to the thrown who were family member of the previous imperial dynasty, and the riot from the people showed an essential vote of no confidence. This shows Republican institutions still held weight in terms of the political dealings during the medieval Roman Empire.

In terms of society and cultural dealings Constantinople and the later Medieval Roman empire was based around a Greek Speaking population. When Rome conquered Greece, Greek culture heavily influenced Roman culture. A Roman aristocrat in the Republican era would have most likely known Greek and studied Greek literature. So while the typical language of republican Rome was Latin they would not been out of place necessarily in the Greek Speaking eastern half of the empire that maintained. Artistically they would have used the same mediums. Fresco's were widely produced and mostly changed from various inspirations of Roman life to more Christian themes, although there remains palace murals depicting wild life of the empire so Christian Themes weren't the only inspiration in Medieval Roman Art.

Architecturally the Romans did not change much other than the scope and use of their buildings. Domes, arches, and pillars were still widely used in the early Medieval period as you can see with the Hagia Sofia.

Cuisine wise things stayed relatively the same but slowly transitioned. Garum was a popular condiment but as time went on the cuisine changed and brought in Eastern elements.

The day to day life of Women remained similar but changed due to Christian influences. Women still retained more legal rights in the medieval times than their contemporaries in the West. They could inherit land, and sign contracts but could no longer lead religious processions. Women were still educated and taught to read and write like the Republic.

So essentially as all things do, culture changed over time but the institutions remained similar while bringing in foreign elements. Language, writing, arts, cuisine, and day to day life remained similar but the contents and inspirations to these aspects changed and morphed over time.